Convolution as addition of supports

Exposition by GPT-6 Astra (OpenAI), 4 October 2026. Public domain (CC0).

Adding two source locations explains both the power and the limitation of convolution. Two compact sources always give a distribution at their sums. Two unbounded sources do so when a bounded output can receive contributions only from a compact set of input pairs. This geometric condition will also control limits, associativity and local regularity.

Pairings are complex-linear, with no conjugation. We use U008, Proposition 1.2 and Theorem 5.1, for finite-order extensions and distributional limits; its formula (T1) characterizes bounded test families. The functional prerequisite, Sections 6 and 14.1–14.2, proves Baire's theorem and completeness of fixed-support test spaces. The scalar prerequisite, Sections 13.1–13.5 and 13.10, and the integration prerequisite, Sections 15.0–15.4, supply differentiation, compact cutoffs, ordinary integration and smooth approximation. All additional distribution constructions used below are proved here. The operator theorem also uses the exact Fourier inversion proof in the Schwartz prerequisite, F2–F4.

Pairings, parameters and tensor products

Write for an open subset of Euclidean space. For tests supported in a fixed compact set, write . A distribution is a linear form with a bound on each such space. A bounded test family has one common compact support and a uniform bound for every , by U008 (T1). Weak convergence means convergence on each test; strong convergence means uniform convergence on every bounded test family. The distinction concerns topologies even though U008 Theorem 5.1 proves that weakly convergent sequences also converge strongly.

B0. Localization and extended pairings. Define as the complement of the union of open sets where vanishes. It is closed, and whenever the compact support of misses it. Here is the localization step behind that assertion. Cover the compact support by finitely many open sets on which vanishes. Choose nonnegative smooth bumps supported in those sets with near the compact support, using smaller balls and the supplied cutoff construction. On that neighborhood divide by , and multiply by a compact cutoff supported there and equal to one near the test support. The smooth functions are extended by zero outside the region where the denominator is positive. They sum to one near the test support. Thus , and every summand pairs to zero. The same partition proves that compatible local distributions glue: evaluate a test by the finite sum of its partitioned local pairings; a common refinement proves independence, and a fixed finite partition near each compact gives continuity.

If and is compact in , set Two choices differ by a compact test whose support misses : at a point in their difference vanishes nearby, and at a point of the function vanishes nearby. B0 therefore proves independence. A common cutoff proves bilinearity whenever the finitely many relevant intersections are compact. Empty intersection gives zero. In particular a compactly supported distribution acts on every smooth function, with an estimate for one fixed cutoff. Multiplication by a smooth function and derivatives, defined by are distributions by the finite product rule and have support contained in that of . A compactly supported distribution consequently has some finite global order. We denote the space of compactly supported distributions by .

A continuous function which defines the zero distribution is zero pointwise. Otherwise multiply its value at a nonzero point by a constant complex phase to make its real part positive; continuity and a nonnegative bump of positive integral in a small neighborhood contradict the zero pairing. This proves uniqueness when smooth local representatives are glued.

B1. Differentiating and integrating parameters. Suppose is smooth and, for in each compact parameter neighborhood, its -support lies in one compact subset of . Then are smooth. Indeed a local order- bound for bounds the change in a pairing by the change in its test. For each , the fundamental theorem gives Uniform continuity on a common compact gives convergence in that seminorm to the derivative, and the same estimate gives continuity of the derivative. Iterate. If has compact support, inserting one cutoff near it gives the statement without a support condition on .

If is compactly supported in both variables, a compact parameter integral can pass through : To prove this, approximate the integral on a containing box by lattice Riemann sums. Every derivative in has uniform Riemann-sum error bounded by the box volume times its modulus of continuity in at the mesh diameter. The sums have a common compact -support. They therefore converge in every test seminorm. Apply the continuity of , and also the scalar Riemann-sum limit to the continuous function in (B2). The same proof works with finitely many continuous -derivatives when these control a finite-order pairing.

B2. Density of product tests and the tensor construction. Finite sums of are dense in , with approximating supports in a fixed compact product. To see this without a tensor theorem, extend a given by zero and form the ordinary integral where both profiles are smooth, nonnegative, compactly supported and normalized. For small , the supports stay inside a fixed compact product in . Differentiating under the averaging integral shows for every . Integration by parts in the ordinary compact integrals identifies these derivatives with derivatives of the kernels. For a fixed , Riemann sums in approximate (B4) in every , by the proof of B1, and each summand is a product test. Choose and then meshes so that the two errors in are at most . This gives one sequence converging in the test topology. The construction can keep supports in any prescribed product neighborhoods of the projections of .

For , , define B1 gives a smooth inner test supported in the first projection of . If is supported in , and the two distributions have orders on those sets, then This proves that it is a distribution. On a product test its value is . Reversing the pairings gives another distribution with those values; B2 proves equality and uniqueness. For three factors, apply B2 first to , and then to each test on . Products of three tests thus determine the distribution. Both associations and every permutation have the same values on them, proving tensor associativity and permutation symmetry. The support is exactly . Outside that product a small rectangle has one zero factor, so B2 and B0 give vanishing. At a point of the product, in every small rectangle each factor has a test with nonzero pairing, and their product has nonzero pairing. The derivative and separate multiplication identities follow either from (B5) and B1 or from their identical values on product tests.

B3. Tensor limits. Tensoring with a fixed distribution is separately weakly and strongly continuous. For weak continuity a fixed gives one fixed inner test by (B5). For strong continuity a bounded family of 's gives a bounded family of inner tests, by B1 and the local estimate for the fixed factor; its support stays in one compact projection.

If and weakly as sequences, U008 Theorem 5.1 supplies common finite orders and constants on any fixed compact projections. Thus (B6) bounds all tensor pairings uniformly in one finite seminorm. Approximate by B2 in that seminorm. On a fixed finite sum of products the pairings converge by ordinary scalar multiplication; the uniform bound controls the approximation error for every and the limit. Consequently The strong conclusion follows from the weak conclusion and U008 Theorem 5.1 applied on the product open set.

Which pairs can contribute to an output

Define translation by For a function this is ; for a point mass it gives .

For closed , addition is proper on the supports if Equivalently, for every finite there is a finite such that The sets are closed, so the equivalence is precisely the closed-and-bounded compactness criterion, applied to containing closed balls. A compact factor implies properness because . Properness implies that is closed: a convergent sequence of sums lies in one compact ball, so its contributing pairs have a convergent subsequence in .

Theorem 1.1 (convolution by addition). If , , and addition is proper on , define The pairing uses B0. It defines a distribution with Orders at most for the two factors give order at most . Also

Proof. The intersection of the tensor support with is closed and contained in the compact set . Hence B0 applies. For all tests supported in a fixed compact , choose one compact smooth on a neighborhood of . Then Estimate (B6), the product rule and the chain rule bound this by a fixed finite seminorm of . When the factor orders are , no term differentiates more than times. Tensor symmetry proves commutativity. A test supported off has zero extended pairing, proving the support assertion.

For one derivative, apply to the compact test in (1.7). The term with vanishes near the tensor support wherever can be nonzero, because on a neighborhood of . Its pairing is zero by B0. The remaining term is , which is the pairing of . Derivatives do not enlarge supports, so the same properness hypothesis permits repetition and differentiation in either variable.

Evaluation at a point in (1.4) and then (1.6) give Thus for every constant-coefficient differential operator , whenever the indicated supports sum properly.

There is also a maximal local output region. Let be the union of open balls such that is compact. For a compact , a finite such cover places , a closed set, inside a finite union of compact sets. Therefore (1.7) defines convolution on . A convergent sequence of sums with limit in eventually lies in one of these balls; compactness of its pairs proves relative closedness of . Consequently Any open region where all compact output sets have compact preimages is contained in , by taking a small closed ball about each point. All preceding binary identities remain valid on this region.

Translating a distribution differentiates with a minus sign

Proposition 1.2 (strong translation derivatives). For every , the map is smooth in the strong distribution topology, and Here smoothness means that all successive parameter difference quotients converge uniformly on each bounded test family, and that their limits are strongly continuous. In particular, for any fixed , The limit is two-sided and strong.

More precisely, fix compact sets of test supports and of translation parameters, and restrict . There are , independent of , such that, for every and integer , The same work for every .

Proof. All tests , with , , have supports in the fixed compact set Choose a finite-order bound for tests supported there. For , repeated use of the fundamental theorem gives For completeness, the case is the fundamental theorem after the substitution . Integration by parts in in the remainder, using , expresses that remainder as plus the next remainder. This proves the formula inductively for both signs of ; is immediate.

Apply (T7) also to every -derivative of order at most . The multinomial expansion, or repeated distributivity of , bounds each derivative of the remainder by The integral is a smooth compactly supported test by the parameter argument B1. Distributional differentiation gives Pairing the remainder with proves (T5).

For continuity under an arbitrary small displacement , the same fundamental-theorem argument on bounds the test difference in by , with one common support set for . On a bounded test family this tends uniformly to zero. Formula (T5) with , divided by , proves the first strong parameter derivative and its sign. Apply this argument successively to every distribution , and use the continuity just proved for those distributions, to obtain (T3). Finally (1.8) identifies , proving (T4). No compact-support assumption on is needed.

For a moving point mass, tends to . For the step function, tends to from either side: for it is the negative interval density , while for it is , also of mass .

Figure. For , the arrows show the coefficients after multiplication by in the first panel and by in the second; they are point masses, not function values. Proposition 1.2 proves the first limit, and Solution 11 proves the second with its quadratic error bound. The Python figure source gives the exact coefficients and labels.

Smoothing and the exact derivative count

Theorem 2.1 (a regular factor). If and , their convolution is the smooth function and The map is continuous from to . If is compactly supported, it is also continuous from to , and from to .

For integers , an order-at-most- distribution convolved with gives a function by (2.1). If has compact support and order at most , the same holds for every .

Proof. On a compact output set , the tests in (2.1) have common support in . B1 proves smoothness and parameter derivatives. Differentiating in reverses the sign of differentiation in , which cancels the distributional derivative sign and proves (2.2). For compact , B0 inserts a fixed cutoff near its support and permits every smooth .

To identify the function with (1.4), pair it with a test . Formula (B3) and the translation substitution give All needed cutoffs can be fixed near the compact contributing projections; B0 then removes them. For supported in a fixed compact , local order of on a compact neighborhood of gives Every defining output seminorm therefore pulls back to an admissible input seminorm in U008's test topology. For compact , its fixed cutoff gives instead a seminorm of on a fixed compact neighborhood of , proving continuity. With fixed compact input support, (1.5) fixes a compact output support as well. The same estimates give continuity on each input support space into that output support space, and hence into by the defining inductive-limit topology.

For finite regularity use U008 Proposition 1.2 to pair with tests. The difference-quotient calculation in B1 converges in whenever the total number of parameter and test derivatives is at most . Uniform continuity of the derivatives of that last order gives continuity of the -th output derivatives. Thus (2.1) is . To justify (2.3) at this regularity, approximate in on a common compact neighborhood by smooth functions. The order bound proves convergence of the function pairings locally uniformly, while the smooth tensor pairings converge distributionally by (B5), or directly by their integral in . This proves agreement with (1.4). For noncompact and compact , apply the same approximation after a cutoff on the only compact region relevant to each output test.

Lemma 2.2 (Riemann sums with derivatives). For , , All sums and the limit have support in .

Proof. The sums are finite. For , compare the differentiated sum with the ordinary integral of on the cubes . The difference between each sampled integrand and its value at a point of that cube is at most These are global moduli of continuity of the zero extensions, so the estimate is uniform in and tends to zero. Only cubes meeting a fixed bounded neighborhood of contribute; their total volume is bounded for small . This proves uniform convergence of every indicated derivative. Differentiating the ordinary integral is justified by the same uniform difference quotients. A nonzero summand requires and ; the integral vanishes off the same compact sum.

Ordinary convolution and the allocation of derivatives

Theorem 2.3 (finite regularity in both factors). If and , then the ordinary convolution is and If in addition , then and Either factor may be the compact one, with the corresponding exchange of their regularities.

Proof. On a compact neighborhood of an output point, all contributing 's in (C1) belong to one compact set , obtained by subtracting from that output neighborhood. The integral is absolutely convergent, bounded by . Uniform continuity of , followed by this bound, proves continuity. Difference quotients of through order converge uniformly on the fixed compact input region to the indicated derivatives, by the ordinary fundamental theorem of calculus. Their integrals therefore converge to the integrals of the derivatives. Induction proves (C1) as actual continuous derivatives.

Substitution gives the equal expression , proving commutation when the two expressions are defined. Apply (C1) first through order , and write its result as . Its integration is on the fixed compact support of . If , difference quotients through order converge uniformly on the compact set of relevant 's. Differentiating in proves (C2), with continuous right-hand side. For any multi-index of length at most , choose of length , and put . All derivatives through that total order exist by this argument, proving . Commutation proves the assertion when the other factor is compact. No boundedness of at infinity was used.

The ordinary integral gives the distributional convolution already constructed. For a compact test , all contributing pairs in lie in one compact product, and the absolute integrand is bounded by a constant times . Fubini and the substitution prove (C3). B5 identifies this with the tensor pairing. When are both compact and continuous, Fubini on the full product also gives For three continuous functions, at least two compact, Theorem 3.1 below gives ordinary associativity: every inner and outer convolution is a continuous function by (C1), their distributions agree by that theorem, and B0 identifies their continuous representatives pointwise.

Associativity and continuity need support control

Theorem 3.1 (associativity). Let be nonempty closed envelopes of the supports of . Suppose that the preimage of every compact set under addition of triples in is compact. Then both iterated convolutions exist and Their common value on is the support-relative pairing of with . In particular, at least two compact factors suffice.

Proof. Fix . Pairs in with sum in a compact , together with third coordinate , lie in the compact triple preimage of . They form a closed subset, so pair addition is proper. Fixing a point of each other nonempty envelope gives the other pair properness statements. The sum is closed by Section 1. For a sequence whose sums lie in a compact set, choose . Triple properness gives a convergent subsequence of . Thus the pairs have a convergent subsequence; their relevant preimage is closed, so addition on is proper. This proves existence of both associations.

Here are the cutoffs that justify equality. Put . Choose a compact smooth cutoff equal to one near the preimage of in . In evaluating , the inner convolution involves only in the compact first projection of . Choose a compact cutoff near all pairs in with sums in that projection, using pair properness. Formula (1.7), B1 and the iterated tensor identity give the compact triple pairing This test is compact: lies in , and in the second projection of . Its cutoff multiplier is one near every contributing point of the triple support with sum in . B0 identifies it with the canonical extended triple pairing. Repeating the same construction for the other association gives the same pairing, by B2's tensor associativity.

If two factors are compact, choose nonempty compact envelopes for them, using for a zero factor, and any nonempty closed envelope for the third. Bounded sums and bounded first two coordinates bound the third. Closedness then proves triple properness.

The nonempty-envelope hypothesis addresses an actual intermediate-existence issue. An empty envelope forces its distribution to be zero. The canonical triple pairing is then zero, and every association whose inner convolution is defined is also defined and zero. It does not define an otherwise missing inner convolution. For example, , , and have a proper empty triple preimage. The association is zero, whereas has no inner convolution under this construction, as (3.5) shows.

For , , , the theorem gives Indeed both are compact distributions. Differentiation under the ordinary compact integral makes smooth. The left side is also a smooth function, by compact integration of the smooth against . The distributional identity therefore holds pointwise by B0. Lemma 2.2 justifies the interchanges even for merely continuous .

When at least one of has compact support, is uniquely characterized by Associativity gives existence. For uniqueness, if a difference has for every test , apply this to the normalized shrinking profiles of Theorem 5.2. That theorem gives , so .

Theorem 3.2 (continuity with fixed support sets). On fixed closed where addition is proper, convolution is separately continuous in both the weak and strong distribution topologies. If , are weakly convergent sequences supported in , then weakly.